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Chapter 3: Problem 45

Find all the real zeros of the polynomial. Use the Quadratic Formula if necessary, as in Example \(3(a)\). $$P(x)=3 x^{3}+5 x^{2}-2 x-4$$

Short Answer

Expert verified
The real zeros are \( x = 1 \), \( x = -\frac{2}{3} \), and \( x = -2 \).

Step by step solution

01

Identify Possible Rational Roots

According to the Rational Root Theorem, possible rational roots of the polynomial \( P(x) = 3x^3 + 5x^2 - 2x - 4 \) are factors of the constant term \(-4\) divided by factors of the leading coefficient \(3\). So, the possible rational roots are \( \pm 1, \pm 2, \pm 4, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3} \).
02

Use Synthetic Division to Test Possible Roots

Use synthetic division to test the possible rational roots. We find that \( x = 1 \) is a root of the polynomial since substituting \( x = 1 \) in for \( P(x) \) gives \( P(1) = 0 \).
03

Divide Polynomial by \( x - 1 \)

Since \( x = 1 \) is a root, divide the polynomial \( P(x) \) by \( x - 1 \) using synthetic division or polynomial division. The quotient is \( 3x^2 + 8x + 4 \).
04

Solve the Quadratic Equation

We now have a quadratic expression \( 3x^2 + 8x + 4 \). Use the Quadratic Formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 3 \), \( b = 8 \), and \( c = 4 \). Calculate the discriminant: \( b^2 - 4ac = 64 - 48 = 16 \).
05

Find the Remaining Zeros

The roots are given by \( x = \frac{-8 \pm \sqrt{16}}{6} \). Solving this gives \( x = \frac{-8 \pm 4}{6} \). The solutions are \( x = \frac{-8 + 4}{6} = \frac{-4}{6} = -\frac{2}{3} \) and \( x = \frac{-8 - 4}{6} = \frac{-12}{6} = -2 \).
06

List All Zeros

The polynomial \( P(x) = 3x^3 + 5x^2 - 2x - 4 \) has the real zeros \( x = 1, -\frac{2}{3}, -2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Synthetic Division
Synthetic Division is a fast and efficient way to divide a polynomial by a simple linear factor of the form \( x - c \). This method reduces the complexity compared to long division. It is particularly beneficial when testing possible roots found using the Rational Root Theorem. Here鈥檚 a simplified guide to using synthetic division:

  • Write down all the coefficients of the polynomial in a row. For our equation, these coefficients are \(3, 5, -2, -4\).
  • Place the chosen root (in this case, \( x = 1 \)) outside the division "box."
  • Bring down the leading coefficient (\(3\) in this instance) to the bottom row.
  • Multiply this number by the root and write the result underneath the next coefficient.
  • Add this result to the next coefficient and write the sum beneath it.
  • Repeat these steps across all coefficients. The last number you obtain is the remainder.
If the remainder is zero, as happened with \( x = 1 \) in this solution, then we've confirmed \( x = 1 \) is a root. This process gives us a quotient polynomial which can be simpler to solve further.
Quadratic Formula
The Quadratic Formula is a key tool for finding the zeros of quadratic equations of the form \( ax^2 + bx + c = 0 \). It provides a straightforward way to solve for the roots without factoring. The formula is:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\\]
This powerful formula allows for straight computation by plugging in the values of \( a \), \( b \), and \( c \).

  • \( a \), \( b \), and \( c \) are coefficients of the quadratic equation.
  • The "\( \pm \)" symbol indicates that there may be two possible solutions.
For the quotient polynomial \( 3x^2 + 8x + 4 \), the application of the Quadratic Formula finds the remaining roots of the original polynomial. Here, after calculating, we have the roots as \( x = -\frac{2}{3} \) and \( x = -2 \).
Discriminant in Quadratic Equations
The discriminant in quadratic equations is the part under the square root in the Quadratic Formula: \( b^2 - 4ac \). It is crucial for determining the nature of the roots of the equation.

  • If the discriminant is positive, there are two distinct real roots.
  • If it equals zero, there is exactly one real double root.
  • If the discriminant is negative, the roots are complex, indicating no real roots exist.

In this exercise, for the polynomial \( 3x^2 + 8x + 4 \), the discriminant calculates to \( 16 \). Since \( 16 \) is positive, we find two distinct real roots, confirming a successful resolution of this segment of the polynomial equation.

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Most popular questions from this chapter

Graph the rational function \(f\), and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. $$f(x)=\frac{x^{3}-2 x^{2}+16}{x-2}, g(x)=x^{2}$$

Use a graphing device to solve the inequality, as in Example 5. Express your answer using interval notation, with the endpoints of the intervals rounded to two decimals. $$x^{3}-2 x^{2}-5 x+6 \geq 0$$

At a certain vineyard it is found that each grape vine produces about 10 lb of grapes in a season when about 700 vines are planted per acre. For each additional vine that is planted, the production of each vine decreases by about 1 percent. So the number of pounds of grapes produced per acre is modeled by $$A(n)=(700+n)(10-0.01 n)$$ where \(n\) is the number of additional vines planted. Find the number of vines that should be planted to maximize grape production.

Graphing Quadratic Functions A quadratic function \(f\) is given. (a) Express \(f\) in standard form. (b) Find the vertex and \(x\) and \(y\) -intercepts of \(f .\) (c) Sketch a graph of \(f .\) (d) Find the domain and range of \(f\). $$f(x)=3 x^{2}+6 x$$

Suppose a rocket is fired upward from the surface of the earth with an initial velocity \(v\) (measured in meters per second). Then the maximum height \(h\) (in meters) reached by the rocket is given by the function $$ h(v)=\frac{R v^{2}}{2 g R-v^{2}} $$ where \(R=6.4 \times 10^{6} \mathrm{m}\) is the radius of the earth and \(g=9.8 \mathrm{m} / \mathrm{s}^{2}\) is the acceleration due to gravity. Use a graphing device to draw a graph of the function \(h .\) (Note that \(h\) and \(v\) must both be positive, so the viewing rectangle need not contain negative values.) What does the vertical asymptote represent physically?
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